In the West, psychiatrists are fully-trained qualified medical doctors, their main difference from psychologists being that they are able to prescibe psychiatric drugs.
The short answer is, psychiatrists are medical doctors and psychologists are not. The suffix "-iatry" means "medical treatment," and "-logy" means "science" or "theory." So psychiatry is the medical treatment of the psyche, and psychology is the science of the psyche.
I thought of psychiatry because sufi Shaykhs "prescribe" treatments - they really are doctors. They don't make you lie down on a couch and answer bizarre questions about your childhood...
NO OFFENCE to any psychologists out there!! I was just drawing a parallel between the approach of Shaykhs and the approach of psychiatrists. Otherwise - I went to several "psychs" to try and cure my crippling insomnia, and neither drugs nor therapy helped. Tasawwuf did!